Some applications of Duhamel product

M. T. Karaev

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 31, Tome 303 (2003), pp. 145-160

Résumé

The Duhamel product of functions $f$ and $g$ is defined by formula $$ (f\circledast g)(x)=\frac{d}{dx}\int^x_0 f(x-t)g(t)\,dt. $$ In the present paper the Duhamel product is used in the study of the spectral multiplicity for direct sums of operators and in the description of cyclic vectors of the restriction of the integration operator in two variables $f(x,y)\mapsto\int^x_0\int^y_0 f(t,\tau)d\tau\,dt$ to its invariant subspace consisting of functions that depend only on the product $xy$.